To simplify the given expression [tex]\(\left( \frac{2x}{7} - \frac{7y}{4} \right)^2\)[/tex], follow these steps:
1. Understand the Expression:
The expression inside the square is a difference of fractions:
[tex]\[
\left( \frac{2x}{7} - \frac{7y}{4} \right)^2
\][/tex]
2. Find a Common Denominator:
To add or subtract fractions, find a common denominator. For [tex]\(\frac{2x}{7}\)[/tex] and [tex]\(\frac{7y}{4}\)[/tex], the common denominator is 28. Rewrite each term with this common denominator:
[tex]\[
\frac{2x}{7} = \frac{2x \cdot 4}{7 \cdot 4} = \frac{8x}{28}
\][/tex]
[tex]\[
\frac{7y}{4} = \frac{7y \cdot 7}{4 \cdot 7} = \frac{49y}{28}
\][/tex]
3. Rewrite the Expression with the Common Denominator:
Now we can rewrite the given expression using the common denominator 28:
[tex]\[
\left( \frac{8x}{28} - \frac{49y}{28} \right)^2
\][/tex]
4. Combine the Fractions:
Since the denominators are now the same, we can combine the terms in the numerator:
[tex]\[
\left( \frac{8x - 49y}{28} \right)^2
\][/tex]
5. Square the Fraction:
To square a fraction, square both the numerator and the denominator separately:
[tex]\[
\left( \frac{8x - 49y}{28} \right)^2 = \frac{(8x - 49y)^2}{28^2}
\][/tex]
6. Simplify the Denominator:
The denominator [tex]\(28^2\)[/tex] is:
[tex]\[
28^2 = 784
\][/tex]
7. Write the Final Simplified Expression:
Therefore, the simplified form of [tex]\(\left( \frac{2x}{7} - \frac{7y}{4} \right)^2\)[/tex] is:
[tex]\[
\frac{(8x - 49y)^2}{784}
\][/tex]
The final simplified answer is:
[tex]\[
\boxed{\frac{(8x - 49y)^2}{784}}
\][/tex]
